Math 225 Linear Algebra II Lecture Notes
John C. Bowman
University of Alberta
Edmonton, Canada
March 23, 2017
, c 2010
John C. Bowman
ALL RIGHTS RESERVED
Reproduction of these lecture notes in any form, in whole or in part, is permitted only for
nonprofit, educational use.
,Contents
1 Vectors 4
2 Linear Equations 6
3 Matrix Algebra 8
4 Determinants 11
5 Eigenvalues and Eigenvectors 13
6 Linear Transformations 16
7 Dimension 17
8 Similarity and Diagonalizability 18
9 Complex Numbers 23
10 Projection Theorem 28
11 Gram-Schmidt Orthonormalization 29
12 QR Factorization 31
13 Least Squares Approximation 32
14 Orthogonal (Unitary) Diagonalizability 34
15 Systems of Differential Equations 40
16 Quadratic Forms 43
17 Vector Spaces: 46
18 Inner Product Spaces 48
19 General linear transformations 51
20 Singular Value Decomposition 53
21 The Pseudoinverse 57
Index 58
, 1 Vectors
• Vectors in Rn :
u1 v1 0
. . .
u = .. , v = .. , 0 = .. .
un vn 0
• Parallelogram law :
u1 + v1
..
u+v = . .
un + vn
• Multiplication by scalar c ∈ R:
cv1
.
cv = .. .
cvn
• Dot (inner ) product:
u·v = u1 v1 + · · · + un vn .
• Length (norm):
√ q
|v|= v·v = v12 + · · · + vn2 .
• Unit vector :
1
v.
|v|
• Distance between (endpoints of) u and v (positioned at 0):
d(u, v) = |u − v|.
• Law of cosines:
|u − v|2 = |u|2 +|v|2 −2|u||v|cos θ.
• Angle θ between u and v:
1 1
|u||v|cos θ = (|u|2 +|v|2 −|u − v|2 ) = (|u|2 +|v|2 −[u − v]·[u − v])
2 2
1
= (|u|2 +|v|2 −[u·u − 2u·v + v·v])
2
= u·v.
4
John C. Bowman
University of Alberta
Edmonton, Canada
March 23, 2017
, c 2010
John C. Bowman
ALL RIGHTS RESERVED
Reproduction of these lecture notes in any form, in whole or in part, is permitted only for
nonprofit, educational use.
,Contents
1 Vectors 4
2 Linear Equations 6
3 Matrix Algebra 8
4 Determinants 11
5 Eigenvalues and Eigenvectors 13
6 Linear Transformations 16
7 Dimension 17
8 Similarity and Diagonalizability 18
9 Complex Numbers 23
10 Projection Theorem 28
11 Gram-Schmidt Orthonormalization 29
12 QR Factorization 31
13 Least Squares Approximation 32
14 Orthogonal (Unitary) Diagonalizability 34
15 Systems of Differential Equations 40
16 Quadratic Forms 43
17 Vector Spaces: 46
18 Inner Product Spaces 48
19 General linear transformations 51
20 Singular Value Decomposition 53
21 The Pseudoinverse 57
Index 58
, 1 Vectors
• Vectors in Rn :
u1 v1 0
. . .
u = .. , v = .. , 0 = .. .
un vn 0
• Parallelogram law :
u1 + v1
..
u+v = . .
un + vn
• Multiplication by scalar c ∈ R:
cv1
.
cv = .. .
cvn
• Dot (inner ) product:
u·v = u1 v1 + · · · + un vn .
• Length (norm):
√ q
|v|= v·v = v12 + · · · + vn2 .
• Unit vector :
1
v.
|v|
• Distance between (endpoints of) u and v (positioned at 0):
d(u, v) = |u − v|.
• Law of cosines:
|u − v|2 = |u|2 +|v|2 −2|u||v|cos θ.
• Angle θ between u and v:
1 1
|u||v|cos θ = (|u|2 +|v|2 −|u − v|2 ) = (|u|2 +|v|2 −[u − v]·[u − v])
2 2
1
= (|u|2 +|v|2 −[u·u − 2u·v + v·v])
2
= u·v.
4
